Grand Unification in the Spectral Pati-Salam Model · Dynamics and interactions If we reconsider the spectral action: Tracee 2=@ M F = 2 ˘ c 4 4Vol(M) + c 0 Z F F 1 j˚j2 2 + j˚j4

Grand Unification in theSpectral Pati-Salam Model

Walter van Suijlekom

Walter van Suijlekom Grand Unification in the Spectral Pati-Salam Model 1 / 32

Contents

• History of the meter: the spectral approach• Fermions in spacetime and emerging bosons• Noncommutative fine structure of spacetime• Examples: electroweak model, Standard Model• Beyond the Standard Model: Pati–Salam unification


History of the meter

Meter defined in 1791 as10−7 times one quarterof the meridian of theEarth.

Expedition in 1792:measuring the arc ofthe meridian betweenBarcelona-Duinkerken,at the beginning of theFrench revolution... a

aAdler (2002)


Meter made concrete by platinum bar “mètre-étalon”,saved (from 1889) in Pavillon de Breteuil near Paris:


Practical objections mètre-étalon (natural variations):

• 1960: meter defined as a multiple of a transition wavelengthin Krypton 86Kr:


• 1967: second = 9192631770 periods of a transition radiationbetween two hyperfine levels in Caesium-133.

• 1983: Definition of the meter as the distance that lighttravels in 1/299792458 second...


So, measuringdistances by look-ing at spectra


A fermion in a spacetime background

Minimal ingredients to describe a free fermion:

• coordinates on spacetime M:

xµ · xν(p) = xµ(p)xν(p), etc .,

with µ, ν = 1, . . . , 4.

• propagation, described by Dirac operator ∂/M = iγµ∂µ, actingon wavefunctions ψ:

S [ψ] =

∫ψ∂/Mψ EOM: ∂/Mψ = 0.

• This combination of coordinate algebra and operators iscentral to the spectral, or noncommutative approach [C 1994].


Emerging bosons

Our fermionic starting point induces a bosonic theory:

• “Inner fluctuations” by the coordinates [C 1996]:

∂/M ∂/M +∑j

aj [∂/M , a′j ]

for functions aj , a′j depending on the coordinates xµ.

• Then, by the chain rule:∑j

aj [∂/M , a′j ] = A

νγµ(∂µxν) = Aµγµ

where Aµ is the electromagnetic 4-potential describing thephoton.


Moreover, it is possible to derive a bosonic action from the(Euclidean) Dirac operator via the spectral action [CC 1996]:

Trace e−∂/2M/Λ

2 ∼ c4Λ4Vol(M) + c2Λ2∫

R√g + c0

∫(∂[µAν])

2 + · · ·

for some coefficients c4, c2, . . ..

We recognize

• The Einstein-Hilbert action∫

R√g for (Euclidean) gravity

• The Lagrangian∫

(∂[µAν])2 for the electromagnetic field


Noncommutative fine structure of spacetime

Replace spacetime byspacetime × noncommutative space: M × F

• F is considered as internal space (Kaluza–Klein like)• F is described by noncommutative matrices, that play the role

of coordinates, just as spacetime is described by xµ(p).

• ‘Propagation’ of particles in F is described by a ’Diracoperator’ ∂/ F which is actually simply a hermitian matrix.

Note that the spectral approach is now the only way to describethe geometry of F .


Example: electroweak theory

Coordinates on F are elements in C⊕H• A complex number z• A quaternion q = q0 + iqkσk ; in terms of Pauli matrices:

σ1 =

(0 11 0

), σ2 =

(0 i−i 0

), σ3 =

(1 00 −1

)It describes a two-point space, with internal structure:

b b

21

z q

Gauge group is given by unitaries: U(1)× SU(2).


’Dirac operator’

∂/ F =

0 c 0c 0 00 0 0

• “Inner fluctuations” can be defined as before but now yield:

∑j

(zj 00 qj

)[∂/ F ,

(z ′j 00 q′j

)]=:

0 cφ1 cφ2cφ1 0 0cφ2 0 0


Almost-commutative spacetimes

∂/M

∂/ F

We combine this mild (matrix) noncommutativity with spacetime:

• coordinates of the almost-commutative spacetime M × F :

x̂µ(p) = (zµ(p), qµ(p))

as elements in C⊕H (for each µ and each point p of M)• The combined Dirac operator becomes

∂/M×F = ∂/M + γ5∂/ F

Note that ∂/ 2M×F = ∂/2M + ∂/

2F , which will be useful later on.


Inner fluctuations on M × FSo, we describe M × F by:

x̂µ = (zµ, qµ) ; ∂/M×F = ∂/M + γ5∂/ F

As before, we consider inner fluctuations of ∂/M×F by x̂µ(p):

• The inner fluctuations of ∂/ F become scalar fields φ1, φ2.• The inner fluctuations of ∂/M become matrix-valued:∑

j

aj [∂/M , a′j ] = aνγ

µ(∂µx̂ν) =: ∂/M + Aµγ

µ

with Aµ taking values in C⊕H:

Aµ =

Bµ 0 00 W 3µ W+µ0 W−µ −W 3µ

corresponding to hypercharge and the W-bosons.


Action functional: electroweak theory

Use ∂/ 2M×F = ∂/2M + ∂/

2F to compute the spectral action

Trace e−∂/2M×F /Λ

2

= Trace e−∂/2M/Λ

2

(1−

∂/ 2FΛ2

+1

2

∂/ 4FΛ4− · · ·

)∼(c4Λ

4Vol(M) + c2Λ2

∫R√g + c0

∫FµνF

µν

)(1−|φ|

2

Λ2+|φ|4

2Λ4

)+· · ·

We now recognize in terms of the field-strength Fµν for Aµ:

• The Yang–Mills term FµνFµν

for hypercharge and W -boson

• The Higgs potential−c4Λ2|φ|2 + 12c4|φ|

4


Standard Model as an almost-commutative spacetime

Describe M × FSM by [CCM 2007]• Coordinates: x̂µ(p) ∈ C⊕H⊕M3(C) (with unimodular

unitaries U(1)Y × SU(2)L × SU(3)).• Dirac operator ∂/M×F = ∂/M + γ5∂/ F where

∂/ F =

(S T ∗

T S

)is a 96× 96-dimensional hermitian matrix where 96 is:

3 × 2 ×( 2⊗ 1 + 1⊗ 1 + 1⊗ 1 + 2⊗ 3 + 1⊗ 3 + 1⊗ 3 )

families

anti-particles

(νL, eL) νR eR (uL, dL) uR dR


The Dirac operator on FSM

∂/ F =

(S T ∗

T S

)

• The operator S is given by

Sl :=

0 0 Yν 00 0 0 YeY ∗ν 0 0 00 Y ∗e 0 0

, Sq ⊗ 13 =

0 0 Yu 00 0 0 YdY ∗u 0 0 00 Y ∗d 0 0

⊗ 13,where Yν , Ye , Yu and Yd are 3× 3 mass matrices acting onthe three generations.

• The symmetric operator T only acts on the right-handed(anti)neutrinos, TνR = YRνR for a 3× 3 symmetric Majoranamass matrix YR , and Tf = 0 for all other fermions f 6= νR .


Inner fluctuations

Just as before, we find

• Inner fluctuations of ∂/M give a matrix

Aµ =

Bµ 0 0 0

0 W 3µ W+µ 0

0 W−µ −W 3µ 00 0 0 (G aµ)

corresponding to hypercharge, weak and strong interaction.

• Inner fluctuations of ∂/ F give(Yν 00 Ye

)

(Yνφ1 −Yeφ2Yνφ2 Yeφ1

)corresponding to SM-Higgs field. Similarly for Yu,Yd .


Dynamics and interactions

If we reconsider the spectral action:

Trace e−∂/2M×F /Λ

2

∼(c4Λ

4Vol(M) + c0

∫FµνF

µν

)(1− |φ|

2

Λ2+|φ|4

2Λ4

)+· · ·

we observe [CCM 2007]:

• The coupling constants of hypercharge, weak and stronginteraction are expressed in terms of the single constant c0which implies

g23 = g22 =

5

3g21

In other words, there should be grand unification.

• Moreover, the quartic Higgs coupling λ is related via

λ ≈ 24 3 + ρ4

(3 + ρ2)2g22 ; ρ =

mνmtop


Phenomenology of the noncommutative Standard Model

This can be used to derive predictions as follows:

• Interpret the spectral action as an effective field theory atΛGUT ≈ 1013 − 1016 GeV.

• Run the quartic coupling constant λ to SM-energies to predict

m2h =4λM2W

3g22

2 4 6 8 10 12 14 16

1.1

1.2

1.3

1.4

1.5

1.6

log10 HΜGeVL

Λ

This gives [CCM 2007]

167 GeV ≤ mh ≤ 176 GeV


Three problems

1 This prediction is falsified by themeasured value.

2 In the Standard Model there isnot the presumed grandunification.

3 There is a problem with the lowvalue of mh, making the Higgsvacuum un/metastable[Elias-Miro et al. 2011].

5 10 15

0.5

0.6

0.7

0.8

log10 HΜGeVL

gaug

eco

uplin

gs

g3

g2

53 g1


Beyond the SM with noncommutative geometryA solution to the above three problems?

• The matrix coordinates of the Standard Model arise naturallyas a restriction of the following coordinates

x̂µ(p) =(qµR(p), q

µL (p),m

µ(p))∈ HR ⊕HL ⊕M4(C)

corresponding to a Pati–Salam unification:

U(1)Y × SU(2)L × SU(3)→ SU(2)R × SU(2)L × SU(4)

• The 96 fermionic degrees of freedom are structured as(νR uiR νL uiLeR diR eL diL

)(i = 1, 2, 3)

• Again the finite Dirac operator is a 96× 96-dimensionalmatrix (details in [CCS 2013]).


Inner fluctuations

• Inner fluctuations of ∂/M now give three gauge bosons:

W µR , WµL , V

µ

corresponding to SU(2)R × SU(2)L × SU(4).• For the inner fluctuations of ∂/ F we distinguish two cases,

depending on the initial form of ∂/ F :

I The Standard Model ∂/ F =

(S T ∗

T S

)II A more general ∂/ F with zero f L − fL-interactions.


Scalar sector of the spectral Pati–Salam model

Case I For a SM ∂/ F , the resulting scalar fields are composite fields,expressed in scalar fields whose representations are:

SU(2)R SU(2)L SU(4)

φbȧ 2 2 1∆ȧI 2 1 4

ΣIJ 1 1 15

Case II For a more general finite Dirac operator, we have fundamentalscalar fields:

particle SU(2)R SU(2)L SU(4)

ΣbJȧJ 2 2 1 + 15

HȧI ḃJ

{31

11

106


Action functional

As for the Standard Model, we can compute the spectral actionwhich describes the usual Pati–Salam model with

• unification of the gauge couplings

gR = gL = g .

• A rather involved, fixed scalar potential, still subject to furtherstudy


Phenomenology of the spectral Pati–Salam model

However, independently from the spectral action, we can analyzethe running at one loop of the gauge couplings [CCS 2015]:

1 We run the Standard Model gauge couplings up to apresumed PS → SM symmetry breaking scale mR

2 We take their values as boundary conditions to thePati–Salam gauge couplings gR , gL, g at this scale via

1

g21=

2

3

1

g2+

1

g2R,

1

g22=

1

g2L,

1

g23=

1

g2,

3 Vary mR in a search for a unification scale Λ where

gR = gL = g

which is where the spectral action is valid as an effectivetheory.


Phenomenology of the spectral Pati–Salam modelCase I: Standard Model ∂/ F

For the Standard Model Dirac operator, we have found that withmR ≈ 4.25× 1013 GeV there is unification at Λ ≈ 2.5× 1015 GeV:


Phenomenology of the spectral Pati–Salam modelCase I: Standard Model ∂/ F

In this case, we can also say something about the scalar particlesthat remain after SSB:

U(1)Y SU(2)L SU(3)(φ01φ+1

)=

(φ1

1̇φ2

1̇

)1 2 1(

φ−2φ02

)=

(φ1

2̇φ2

2̇

)−1 2 1

σ 0 1 1

η −23

1 3

• It turns out that these scalar fields have a little influence onthe running of the SM-gauge couplings (at one loop).

• However, this sector contains the real scalar singlet σ thatallowed for a realistic Higgs mass and that stabilizes the Higgsvacuum [CC 2012].


Phenomenology of the spectral Pati–Salam modelCase II: General Dirac

For the more general case, we have found that withmR ≈ 1.5× 1011 GeV there is unification at Λ ≈ 6.3× 1016 GeV:


Conclusion

With our walk through the noncommutative garden, we havearrived at a spectral Pati–Salam model that

• goes beyond the Standard Model• has a fixed scalar sector once the finite Dirac operator has

been fixed (only a few scenarios)

• exhibits grand unification for all of these scenarios (confirmedby [Aydemir–Minic–Sun–Takeuchi 2015])

• the scalar sector has the potential to stabilize the Higgsvacuum and allow for a realistic Higgs mass.


Further reading

A. Chamseddine, A. Connes, WvS.

Beyond the Spectral Standard Model: Emergence ofPati-Salam Unification. JHEP 11 (2013) 132.[arXiv:1304.8050]

Grand Unification in the Spectral Pati-Salam Model. Toappear in JHEP [arXiv:1507.08161]

WvS.

Noncommutative Geometry and Particle Physics.Mathematical Physics Studies, Springer, 2015.

and also: http://www.noncommutativegeometry.nl


Documents

Grand Unification in the Spectral Pati-Salam Model · Dynamics and interactions If we reconsider the spectral action: Tracee 2=@ M F = 2 ˘ c 4 4Vol(M) + c 0 Z F F 1 j˚j2 2 + j˚j4